Balanced fiber bundles and GKM theory

2011/10/18 by Guillemin, Victor, Sabatini, Silvia, Zara, Catalin
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1110.4086

Abstract

Let T be a torus and B a compact T-manifold. Goresky, Kottwitz, and MacPherson show in \citeGKM that if B is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant cohomology ring H_T^*(B) as a subring of H_T^*(B^T). In this paper we prove an analogue of this result for T-equivariant fiber bundles: we show that if M is a T-manifold and π\colon M → B a fiber bundle for which π intertwines the two T-actions, there is a simple combinatorial description of H_T^*(M) as a subring of H_T^*(π^-1(B^T)). Using this result we obtain fiber bundle analogues of results of \citeGHZ on GKM theory for homogeneous spaces.

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